Interpolatory Weighted-H2 Model Reduction

This paper introduces an interpolation framework for the weighted-H2 model reduction problem. We obtain a new representation of the weighted-H2 norm of SISO systems that provides new interpolatory first order necessary conditions for an optimal reduced-order model. The H2 norm representation also provides an error expression that motivates a new weighted-H2 model reduction algorithm. Several numerical examples illustrate the effectiveness of the proposed approach

On the Scalar Rational Interpolation Problem

The rational interpolation problem in the scalar case, including multiple points, is solved. In particular a parametrization of all minimal-degree rational functions interpolating given pairs of points is derived. These considerations provide a generalization of the results on the partial realization of linear system

Case study: Approximations of the Bessel Function

The purpose of this note is to compare various approximation methods as applied to the inverse of the Bessel function of the first kind, in a given domain of the complex plane.Comment: 18 pages, 44 figure

Data-driven quadratic modeling in the Loewner framework

In this work, we present a non-intrusive, i.e., purely data-driven method that uses the Loewner framework (LF) along with nonlinear optimization techniques to identify or reduce quadratic control systems from input-output (i/o) time-domain measurements. At the heart of this method are optimization schemes that enforce interpolation of the symmetric generalized frequency response functions (GFRFs) as derived in the Volterra series framework. We consider harmonic input excitations to infer such measurements. After reaching the steady-state profile, the symmetric GFRFs can be measured from the Fourier spectrum (phase and amplitude). By properly using these measurements, we can identify low-order nonlinear state-space models with non-trivial equilibrium state points in the quadratic form, such as the Lorenz attractor. In particular, for the multi-point equilibrium case, where measurements describe some local bifurcated models to different coordinates, we achieve global model identification after solving an operator alignment problem based on a constrained quadratic matrix equation. We test the new method for a more demanding system in terms of state dimension, i.e., the viscous Burgers' equation with Robin boundary conditions. The complexity reduction and approximation accuracy are tested. Future directions and challenges conclude this work.Comment: 29 pages, 7 figure

Density Matrix Renormalization for Model Reduction in Nonlinear Dynamics

We present a novel approach for model reduction of nonlinear dynamical systems based on proper orthogonal decomposition (POD). Our method, derived from Density Matrix Renormalization Group (DMRG), provides a significant reduction in computational effort for the calculation of the reduced system, compared to a POD. The efficiency of the algorithm is tested on the one dimensional Burgers equations and a one dimensional equation of the Fisher type as nonlinear model systems.Comment: 12 pages, 12 figure

When is the discretization of a spatially distributed system good enough for control?

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